Consider a population proportion p = 0.27. [You may find it useful to reference the z table.]
a. What is the expected value and the standard error of the sampling distribution of the sample proportion with n = 17 and n = 65? (Round the standard error to 4 decimal places.)
b. Can you conclude that the sampling distribution of the sample proportion is normally distributed for both sample sizes?
multiple choice
Yes, the sampling distribution of the sample proportion is normally distributed for both sample sizes.
No, the sampling distribution of the sample proportion is not normally distributed for either sample size.
No, only the sample proportion with n = 17 will have a normal distribution.
No, only the sample proportion with n = 65 will have a normal distribution.
c. If the sampling distribution of the sample proportion is normally distributed with n = 17, then calculate the probability that the sample proportion is between 0.25 and 0.27. (If appropriate, round final answer to 4 decimal places.)
d. If the sampling distribution of the sample proportion is normally distributed with n = 65, then calculate the probability that the sample proportion is between 0.25 and 0.27. (If appropriate, round final answer to 4 decimal places.)
a.
For n = 17:
Expected Value = p = 0.27
Standard Error = sqrt(p(1-p)/n) = sqrt(0.27(1-0.27)/17) ≈ 0.0624
For n = 65:
Expected Value = p = 0.27
Standard Error = sqrt(p(1-p)/n) = sqrt(0.27(1-0.27)/65) ≈ 0.0480
b. No, only the sample proportion with n = 65 will have a normal distribution.
c. To calculate the probability that the sample proportion is between 0.25 and 0.27 with n = 17, we can calculate the z-scores for the lower and upper bounds and use the z-table.
Lower Bound z-score = (0.25 - 0.27) / 0.0624 ≈ -0.0321
Upper Bound z-score = (0.27 - 0.27) / 0.0624 = 0.0000
Using the z-table, we find the probability associated with the lower bound z-score is approximately 0.4861 and the probability associated with the upper bound z-score is approximately 0.5000.
Therefore, the probability that the sample proportion is between 0.25 and 0.27 with n = 17 is approximately 0.5000 - 0.4861 = 0.0139.
d. To calculate the probability that the sample proportion is between 0.25 and 0.27 with n = 65, we can repeat the same steps as in part c.
Lower Bound z-score = (0.25 - 0.27) / 0.0480 ≈ -0.4167
Upper Bound z-score = (0.27 - 0.27) / 0.0480 = 0.0000
Using the z-table, we find the probability associated with the lower bound z-score is approximately 0.3372 and the probability associated with the upper bound z-score is approximately 0.5000.
Therefore, the probability that the sample proportion is between 0.25 and 0.27 with n = 65 is approximately 0.5000 - 0.3372 = 0.1628.
a. To find the expected value and standard error of the sampling distribution of the sample proportion, we can use the formulas:
Expected Value (mean):
E(p) = p
Standard Error:
SE(p) = sqrt((p * (1 - p)) / n)
For n = 17:
E(p) = 0.27
SE(p) = sqrt((0.27 * (1 - 0.27)) / 17) = sqrt(0.1851 / 17) ≈ 0.1441
For n = 65:
E(p) = 0.27
SE(p) = sqrt((0.27 * (1 - 0.27)) / 65) = sqrt(0.1851 / 65) ≈ 0.0721
b. The sampling distribution of the sample proportion is approximately normally distributed when the sample size is large enough (n > 30) due to the Central Limit Theorem. Both n = 17 and n = 65 are above this threshold, so the correct answer is:
Yes, the sampling distribution of the sample proportion is normally distributed for both sample sizes.
c. To calculate the probability that the sample proportion is between 0.25 and 0.27, we need to find the z-scores for both values and use the z-table.
For n = 17:
Z1 = (0.25 - 0.27) / 0.1441 ≈ -0.0139
Z2 = (0.27 - 0.27) / 0.1441 = 0
Using the z-table, we can find the area to the left of Z1 and subtract it from the area to the left of Z2 to find the probability. Let's denote the area to the left of a z-score as P(Z < z). Using the table or a calculator, we find that P(Z < -0.0139) ≈ 0.4956 and P(Z < 0) = 0.5.
Therefore, the probability that the sample proportion is between 0.25 and 0.27 when n = 17 is approximately:
P(0.25 < p < 0.27) ≈ P(Z < 0) - P(Z < -0.0139) ≈ 0.5 - 0.4956 ≈ 0.0044
d. To calculate the probability that the sample proportion is between 0.25 and 0.27 when n = 65, we follow the same steps as in part c:
Z1 = (0.25 - 0.27) / 0.0721 ≈ -0.0278
Z2 = (0.27 - 0.27) / 0.0721 = 0
Using the z-table, we find P(Z < -0.0278) ≈ 0.4896 and P(Z < 0) = 0.5.
Therefore, the probability that the sample proportion is between 0.25 and 0.27 when n = 65 is approximately:
P(0.25 < p < 0.27) ≈ P(Z < 0) - P(Z < -0.0278) ≈ 0.5 - 0.4896 ≈ 0.0104
a. To find the expected value and the standard error of the sampling distribution of the sample proportion, we can use the formulas:
Expected Value (μ) = p
Standard Error (SE) = sqrt((p*(1-p))/n)
For n = 17:
μ = 0.27
SE = sqrt((0.27*(1-0.27))/17) ≈ 0.0632 (rounded to 4 decimal places)
For n = 65:
μ = 0.27
SE = sqrt((0.27*(1-0.27))/65) ≈ 0.0426 (rounded to 4 decimal places)
b. To determine if the sampling distribution of the sample proportion is normally distributed for both sample sizes, we can check if the conditions for the Central Limit Theorem are met. The conditions are:
1. Randomization: The sample must be a random sample or sufficiently close to one. (Assuming this condition is met.)
2. Independence: Observations must be independent of each other. (Assuming this condition is met.)
3. Sample Size: The sample size must be sufficiently large, typically at least 30.
For n = 17, the sample size is less than 30, so the Central Limit Theorem may not apply, and the sampling distribution may not be perfectly normal.
For n = 65, the sample size is greater than 30, so the Central Limit Theorem is more likely to apply, and the sampling distribution is more likely to be approximately normal.
Therefore, the correct answer is "No, only the sample proportion with n = 65 will have a normal distribution."
c. To calculate the probability that the sample proportion is between 0.25 and 0.27 when the sampling distribution is normally distributed with n = 17, we can use the z-table.
First, we need to convert the values into z-scores using the formula:
z = (x - μ) / SE
For 0.25:
z1 = (0.25 - 0.27) / 0.0632 ≈ -0.3165
For 0.27:
z2 = (0.27 - 0.27) / 0.0632 = 0
Next, we look up the corresponding probabilities in the z-table. The probability that the z-score falls between -0.3165 and 0 is approximately 0.3757.
Therefore, the probability that the sample proportion is between 0.25 and 0.27 when the sampling distribution is normally distributed with n = 17 is approximately 0.3757 (rounded to 4 decimal places).
d. Using the same process as in part c, but with n = 65, we can calculate the probability that the sample proportion is between 0.25 and 0.27.
For 0.25:
z1 = (0.25 - 0.27) / 0.0426 ≈ -0.4695
For 0.27:
z2 = (0.27 - 0.27) / 0.0426 = 0
Looking up the corresponding probabilities in the z-table, the probability that the z-score falls between -0.4695 and 0 is approximately 0.3192.
Therefore, the probability that the sample proportion is between 0.25 and 0.27 when the sampling distribution is normally distributed with n = 65 is approximately 0.3192 (rounded to 4 decimal places).